Filtering the Navier-Stokes Equation

Transcription

1 Filtering the Navier-Stokes Equation Andrew M Stuart1 1 Mathematics Institute and Centre for Scientific Computing University of Warwick Geometric Methods Brown, November 4th 11 Collaboration with C. Brett, A. Lam, K.J.H. Law, D. McCormick and M. Scott (Warwick). Funded by EPSRC, ERC and ONR

2 Outline 1 3 4

3 Outline 1 3 4

4 Forward Problem: Navier Stokes Equations Let f H. D Navier-Stokes as ODE on H : dv + νav + F (v ) = f, dt v () = u Here Z o n u dx =, norm H = u L (T ) u =, ZT n o V = u H 1 (T ) u =, u dx =, norm k k T Introduce semigroup notation in H (weak) or V (strong): v (t) = Ψ(u; t), Ψ(u) = Ψ(u; h), vj = Ψ(u; jh).

5 Forward Problem: Navier Stokes Equations Let f H. D Navier-Stokes as ODE on H : dv + νav + F (v ) = f, dt v () = u Here Z o n u dx =, norm H = u L (T ) u =, ZT n o V = u H 1 (T ) u =, u dx =, norm k k T Introduce semigroup notation in H (weak) or V (strong): v (t) = Ψ(u; t), Ψ(u) = Ψ(u; h), vj = Ψ(u; jh).

6 Forward Problem: Navier Stokes Equations Let f H. D Navier-Stokes as ODE on H : dv + νav + F (v ) = f, dt v () = u Here Z o n u dx =, norm H = u L (T ) u =, ZT n o V = u H 1 (T ) u =, u dx =, norm k k T Introduce semigroup notation in H (weak) or V (strong): v (t) = Ψ(u; t), Ψ(u) = Ψ(u; h), vj = Ψ(u; jh).

7 Inverse Problem: Navier Stokes Equations Orthogonal projections onto low and high divergence-free Fourier modes ϕk (eigenfunctions of A): Pλ : H 7 {ϕk (x), k λ}, Qλ : H 7 {ϕk (x), k > λ.} Observations: yj = Pλ vj + ξj, Yj = {yi }ji=1. Goal: find vj given data Yj. ξj N(, Γ)

8 Inverse Problem: Navier Stokes Equations Orthogonal projections onto low and high divergence-free Fourier modes ϕk (eigenfunctions of A): Pλ : H 7 {ϕk (x), k λ}, Qλ : H 7 {ϕk (x), k > λ.} Observations: yj = Pλ vj + ξj, Yj = {yi }ji=1. Goal: find vj given data Yj. ξj N(, Γ)

9 Inverse Problem: Navier Stokes Equations Orthogonal projections onto low and high divergence-free Fourier modes ϕk (eigenfunctions of A): Pλ : H 7 {ϕk (x), k λ}, Qλ : H 7 {ϕk (x), k > λ.} Observations: yj = Pλ vj + ξj, Yj = {yi }ji=1. Goal: find vj given data Yj. ξj N(, Γ)

10 3DVAR: Approximate Gaussian Filter Impose the (3DVAR) Gaussian approximation: b j, Cb P(vj Yj ) N m b j ), C. P(vj+1 Yj ) N Ψ(m Kalman Mean Update: b j+1 = BΨ(m b j ) + (I B)yj+1. m Kalman Covariance Update: Cb 1 = C 1 + Pλ Γ 1 Pλ, b 1. B = CC

11 3DVAR: Approximate Gaussian Filter Impose the (3DVAR) Gaussian approximation: b j, Cb P(vj Yj ) N m b j ), C. P(vj+1 Yj ) N Ψ(m Kalman Mean Update: b j+1 = BΨ(m b j ) + (I B)yj+1. m Kalman Covariance Update: Cb 1 = C 1 + Pλ Γ 1 Pλ, b 1. B = CC

12 3DVAR: Approximate Gaussian Filter Impose the (3DVAR) Gaussian approximation: b j, Cb P(vj Yj ) N m b j ), C. P(vj+1 Yj ) N Ψ(m Kalman Mean Update: b j+1 = BΨ(m b j ) + (I B)yj+1. m Kalman Covariance Update: Cb 1 = C 1 + Pλ Γ 1 Pλ, b 1. B = CC

13 Unstable 3DVAR, ν=.1, h=. 3DVAR, ν=.1, h=., Re(u1,) m.3 1 m(tn) u+(tn) tr(γ) tr[(i Bn)Γ(I Bn)*] 1 1 u+ yn step 1 3 t 4

14 Stabilized [3DVAR], ν=.1, h=., Re(u1,) [3DVAR], ν=.1, h=. m(tn) u+(tn) tr(γ) tr[(i Bn)Γ(I Bn)*] 1 m.3 u+ yn step t 1 1

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16 Exemplar for Theorem C = δ A ζ (Model covariance); Γ = σ A β (Observation covariance); define η = σ /δ, α = ζ β; η is ratio of observation variance to model variance. b j+1 = BΨ(m b j ) + (I B)yj+1 m B = (I + η Aα ) 1 η Aα, B = I, on Qλ H. on Pλ H,

17 Stability Theorem Recall vj = Ψ(j) (u) = v (jh). Define supj 1 kξj k =. How does error in filter mean behave? Theorem (BLLMSS11) Assume that: b BV (, R); m λ sufficiently large and h sufficiently small. Then ηc = ηc (R) and r (, 1), c (, ) such that, for all η < ηc, b j vj k r j km b v k + c. km

18 Proof of Stability Theorem vj+1 = Ψ(vj ) vj+1 = BΨ(vj ) + (I B)Ψ(vj ) b j+1 = BΨ(m b j ) + (I B)yj+1 m b j+1 = BΨ(m b j ) + (I B)Ψ(vj ) + (I B)ξj. m ej+1 BDΨ vj )ej + (I B)ξj.

19 Proof of Stability Theorem vj+1 = Ψ(vj ) vj+1 = BΨ(vj ) + (I B)Ψ(vj ) b j+1 = BΨ(m b j ) + (I B)yj+1 m b j+1 = BΨ(m b j ) + (I B)Ψ(vj ) + (I B)ξj. m ej+1 BDΨ vj )ej + (I B)ξj.

20 Proof of Stability Theorem vj+1 = Ψ(vj ) vj+1 = BΨ(vj ) + (I B)Ψ(vj ) b j+1 = BΨ(m b j ) + (I B)yj+1 m b j+1 = BΨ(m b j ) + (I B)Ψ(vj ) + (I B)ξj. m ej+1 BDΨ vj )ej + (I B)ξj.

21 Proof of Stability Theorem (Continued) ej+1 BDΨ vj )ej + (I B)ξj. BDΨ( ) can be made a contraction on Qλ H by choice of λ; key estimate from Hayden, Olsen and Titi, PhysicaD, 11 BDΨ( ) can be made a contraction on Pλ H by choice of η sufficiently small: variance inflation.

22 Proof of Stability Theorem (Continued) ej+1 BDΨ vj )ej + (I B)ξj. BDΨ( ) can be made a contraction on Qλ H by choice of λ; key estimate from Hayden, Olsen and Titi, PhysicaD, 11 BDΨ( ) can be made a contraction on Pλ H by choice of η sufficiently small: variance inflation.

23 Proof of Stability Theorem (Continued) ej+1 BDΨ vj )ej + (I B)ξj. BDΨ( ) can be made a contraction on Qλ H by choice of λ; key estimate from Hayden, Olsen and Titi, PhysicaD, 11 BDΨ( ) can be made a contraction on Pλ H by choice of η sufficiently small: variance inflation.

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25 The Context η is ratio of observation variance to model variance. Previous Theorem required η 1. Here we study the limit η 1. High frequency observation compensates.

26 Parameter Scalings C = ωσ A ζ ; Γ = h1 σ A β ; yj := zj+1 zj h ; yj = Pλ vj + ξj ; ξj σ N(, A β ). h

27 SPDE Limit Formal diffusion limit: b dm dz b + F (m) b + ωa α m b + νam = f, dt dt where the data z solves dz = vdt + σ A β dw, z() =. b b. m() =m

28 SPDE Stability u1, 1 m1, u14,.1.5 m14,.1.5 t 4 6 t u7,.1 m7, 4 6 m(t) u(t) / u(t) t 4 6 t 4 6

29 SPDE Stability 1. u1, m1, t 15. m14, u7, m7,.1 u14,.1 1 t 15 m(t) u(t) / u(t) t t 15

30 Outline 1 3 4

31 Conclusions Approximate filters are routinely used in geophysical applications. They fail to reproduce covariance but can accurately track the mean. They can exhibit instability on longer time-intervals. Variance inflation provably stabilizes. SPDE in high frequency limit. Future work: ExKF, EnKF.

32 Conclusions Approximate filters are routinely used in geophysical applications. They fail to reproduce covariance but can accurately track the mean. They can exhibit instability on longer time-intervals. Variance inflation provably stabilizes. SPDE in high frequency limit. Future work: ExKF, EnKF.

33 Conclusions Approximate filters are routinely used in geophysical applications. They fail to reproduce covariance but can accurately track the mean. They can exhibit instability on longer time-intervals. Variance inflation provably stabilizes. SPDE in high frequency limit. Future work: ExKF, EnKF.

34 Conclusions Approximate filters are routinely used in geophysical applications. They fail to reproduce covariance but can accurately track the mean. They can exhibit instability on longer time-intervals. Variance inflation provably stabilizes. SPDE in high frequency limit. Future work: ExKF, EnKF.

35 Conclusions Approximate filters are routinely used in geophysical applications. They fail to reproduce covariance but can accurately track the mean. They can exhibit instability on longer time-intervals. Variance inflation provably stabilizes. SPDE in high frequency limit. Future work: ExKF, EnKF.

36 Conclusions Approximate filters are routinely used in geophysical applications. They fail to reproduce covariance but can accurately track the mean. They can exhibit instability on longer time-intervals. Variance inflation provably stabilizes. SPDE in high frequency limit. Future work: ExKF, EnKF.

37 http : // / masdr [CDRS9]: S.L. Cotter, M. Dashti and A.M.Stuart. "Bayesian inverse problems for functions and applications to fluid mechanics". Inverse Problems 5 (9) [LS11]: K.J.H.Law and A.M.Stuart. "Evaluating Data Assimilation Algorithms". [BLLMSS11]: C. Brett, A. Lam, K.J.H. Law, D. McCormick, M. Scott and A.M. Stuart. "Stability of Filters for the Navier-Stokes Equation. Bayesian Overview: A.M. Stuart "Inverse problems: a Bayesian perspective. Acta Numerica 19 (1), Data Assimilation Review: A. Apte, C.K.R.T. Jones, A.M. Stuart and J. Voss, Data assimilation: mathematical and statistical perspectives. Int. J. Num. Meth. Fluids 56(8),

38 General References 4DVAR: Bennett A. "Inverse modeling of the ocean and atmosphere." Cambridge. (). 3DVAR: Kalnay E. "Atmospheric modeling, data assimilation, and predictability." Cambridge Univ Pr. (3). FDF Majda A, Harlim J, Gershgorin B. "Mathematical strategies for filtering turbulent dynamical systems." Dynamical Systems 7(): (1) ExKF Jazwinski A. "Stochastic processes and filtering theory." Academic Pr. (197). EnKF Evensen G. "Data assimilation: the ensemble kalman filter." Springer Verlag. (9). Geophysical Applications Van Leeuwen, P.J. "Particle filtering in geophysical systems." Monthly Weather Review 137(9),